1- Find the volume of the solid obtained from rotating the region enclosed by th

Question

1- Find the volume of the solid obtained from rotating the region enclosed by the curves y = 15 - x, y = 2x + 9, x = -1 about the x-axis. (Hint: Sketch the graph.)

2- Find the volume of the solid obtained from rotating the region enclosed by the graphs x = 2 - y, x = 4 - y2 about the y-axis for -1 %u2264 y %u2264 2.

3- Find the volume of the solid obtained by rotating region A in the figure below about x = -3.


Find the volume of the solid obtained by rotating region A in the figure below about y = 0.

4- Find the volume of the solid obtained by rotating the region in the first quadrant enclosed by the graphs y = x2, y = 2 - x, x = 0, about

Explanation / Answer

Since you are finding it about the y-axis, you must solve the equations in terms of y.

y = 5x and y = (1/5)x^2
x = y/5 and x = %u221A(5y)

Now determine which graph is more rightwards. I used my graphing calculator to find that y = (1/5)x^2 is more rightwards. When doing the Washer Method about a vertical line, you do right volume - left volume.

V = %u03C0 %u222B [(%u221A(5y)^2) - ((y/5)^2)] dy
V = %u03C0 %u222B [(5y) - (y^2/25)] dy
V = %u03C0 [5y^2/2 - y^3/75]

To determine the limits of integration, you must find the y values at which the graphs intersect. You can do so by setting the functions equal to each other: y/5 = %u221A(5y). In doing so, you find that they intersect at y=0 and at y=125. These are your limits of integration.

Now,

V = %u03C0 [5y^2/2 - y^3/75] from 0 to 125
V = %u03C0 ([39062.5] - [26041.6667])
V = 40906.15

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